Section 1.8 Diffraction Gratings
A diffraction grating consists of many very small, equally spaced slits. A grating can be either a transmission grating (as pictured in Figure 1.41, where the light comes from one side and gets diffracted out the other, or a reflection grating, where the light gets diffracted back on the same side of the grating that it came from. The “slits” may consist of absorbing and transparent regions, as with our double and triple slits, or simply of ridges in a transparent material, or of angled reflecting surfaces. Historically, the first diffraction gratings were made by painstakingly etching thousands of microscopic parallel grooves in glass using a diamond scribe. Fortunately, the details of how the grooves, lines, or slits are shaped doesn't affect where the light gets diffracted. (It does affect how much light gets diffracted, but we won't worry about calculating that.)
Everything we did for two and three slits carries over to a diffraction grating, except that now we have a very large number of phasors (in the range of 1000 to 10,000). That means we can't easily draw them all and add them as vectors, but we can still figure out what the diffraction pattern must look like.
Suppose our grating is illuminated at normal incidence by light of wavelength \(\lambda\text{,}\) so that the light reaches all the slits in phase. Each slit re-radiates its own outgoing wave, and these waves interfere on a distant screen. We represent each interfering wave as a phasor on a phasor diagram. To get a maximum from our many-phasor diagram, we want all the phasors pointing in the same direction, as in Figure 1.42. This happens when the phase difference between two adjacent phasors is \(0\text{,}\) but also when it is \(2\pi\text{,}\) \(4\pi\text{,}\) etc., and when it is \(-2\pi\text{,}\) \(-4\pi\text{,}\) etc. Thus we have maxima in the intensity for any integer multiple of \(2\pi\text{,}\) provided the angle of the outgoing light is small enough that the light can actually reach the screen. These maxima are often referred to as “orders” of diffraction, where \(\Delta\phi=0\) is called the zeroth order, \(\Delta\phi=2\pi\) is the first order, \(\Delta\phi = 4\pi\) is the second order, \(\Delta\phi=-2\pi\) the minus-first order, etc.
What about points on the screen that aren't maxima? If we move away from the central maximum by a small distance, there's now some non-zero phase difference \(\Delta\phi_\text{ adj. }\text{.}\) This means that each adjacent phasor is rotated by \(\Delta\phi_\text{ adj. }\) relative to its neighbor. But remember, we now have A LOT of phasors. So the by the time we've gotten through our 1000 or 10,000 phasors, the phasor diagram has wrapped itself up in a tight little circles, as indicated in Figure 1.43. The amplitude won't be exactly zero, but it will be much less than the amplitude at the maximum. And the intensity, which is proportional to amplitude squared, will be MUCH, MUCH less than at the maximum. This is why the diffraction pattern from the grating consists of very narrow maxima, where all the light is concentrated, and wide expanses in between where essentially no light goes. (Incidentally, you can see from conservation of energy that when the maxima get very bright, they must also get very narrow–otherwise the total light power reaching the screen would be bigger than the light power passing through the slit!)
You can think of the grating diffraction pattern as the limit of the \(N\)-slit pattern, when \(N\) gets big. As shown in Figure 1.44, the primary maxima of the multi-slit pattern get narrower and narrower as the number of slits increases (assuming a constant slit spacing). These maxima get correspondingly brighter, since if the amplitude from each slit is \(A\text{,}\) the combined amplitude from \(N\) slits at a maximum is \(N A\text{,}\) and the combined intensity is proportional to \(N^2 A^2\text{,}\) which grows rapidly as the number of slits increases.
